Statistics 410  Introduction to Probability

Fall 2019 MW 5-6:15pm,    Mth 0303

Instructor: Eric Slud, Statistics program, Math. Dept.

Office:  MTH 2314, x5-5469, email evs@math.umd.edu, Office Hours: M11, W10, or by appointment

Please fill out the on-line Evaluation form on this Course and instructor, by Tuesday Dec. 10, at http://CourseEvalUM.umd.edu. Thank you.

Also please note that I am scheduling a review session for the Final Exam, for anyone who wants it, on the afternoon of Wednesday December 11 4:30-6pm, as an extended office hour in my office at MTH 2314.

Sample problems for the Final Exam (with solutions) are posted here. Unlike the in-class Tests, at the Final Exam you will be allowed to use a memory-aid (cheat-sheet) consisting of at most two 8.5 x 11" notebook-sheet sides.

Course Text: Sheldon Ross, A First Course in Probability, 10th ed., Pearson. We will cover Chapters 1-8, and some of Chapter 10 on Simulation as time permits.

Current Homework Assignment

Syllabus: This semester's syllabus, based on the Ross 10th edition.

Overview: This course is a solid introduction to the formulation and manipulation of probability models, leading up to proofs of limit theorems: the law of large numbers and the central limit theorem. It is a gateway course to serious study of mathematical statistics and graduate-level applied statistics. Key topics characterizing this course as opposed to more elementary introductions to Probability include joint distributions and change-of-variable formulas for them; conditional expectation and its applications; and the formal proofs of limit theorems.

Prerequisite: Math 240-241 or Math 340-341.

Course requirements and Grading: there will be graded homework sets (one every 2 weeks, 5 altogether) which together will count 20% of the course grade; (two or three) 15-minute quizzes in-class that will together count 10%; 2 tests that will count 20% each; and a final exam that will count 30%.

Homework

Notes and Guidelines. (a) Homeworks should be handed in as hard-copy in-class, except for possible occasional due-dates on Fridays when you may submit them electronically, via email, in pdf format. A percentage deduction (at least 15%) of the overall HW score will generally be made for late papers.
(b) Some of the problems will have answers in the back of the book, and some will be similar to problems that can be found in the 9th or earlier editions of the book. You may of course use such information as aids, but your submitted HW solutions must show all work, self-contained, to get full credit.

Homework 1 (due Friday 9/6, 5pm):    Reading is all of Chapter 1 of 10th edition.
Problems, pp.16-17: #8 (b)-(c), 10, 18.
Theoretical Problems, pp. 18-19: #8, 11.
Also: calculate how many poker hands there are with (a) 1 pair (i.e. pattern xxyzw of card values), (b) three of a kind (pattern xxxyz), and (c) 2 pair (pattern xxyyz).

Homework 2 (due Wednesday 9/25, 5pm; 11 problems in all):    Reading consists of Chapters 2, 3 and 4 through 4.8.3.
Ch.2 Problems, pp.50-54: #7, #27, #32. [But in #27, first find the requested probability and then also find the probability of the event that A is the first to select the red ball and that this happens at A's 3rd pick.]
Ch.3 Problems, pp.103-112: #19, #23, #54. Note that we are not assuming anything about any player's winning a game other than player A. But we are assuming that the events of winning separate games are all (jointly) independent.
Ch.4 Problems, pp.175-179: #7, 48.

Additional problems (also part of the required 10 to be handed in):
(I). Find the probability that a 5-card poker hand from a carefully shuffled deck,
(a) contains 2 red cards and is a 3-of-a-kind hand (i.e., the card-values have the pattern xxxyz),
(b) contains 3 spades no duplicated card-values conditionally given that it has no red-suit (Heart or Diamond) cards in it.
(II). In terms of the probabilities of three events A, B, C and their 2- or 3-way intersections (e.g., A∩B, A∩B∩C) give a general expression for the probability that exactly one of these events A, B, or C occurs.
(III). A fair die is tossed repeatedly and independently according to the following rules: at the first toss where a 1, 2 or 4 is seen, the game stops, the number X of the toss is recorded, and the number Y is defined equal to 2*X if the final toss was a 4, and Y is defined as X/2 if the final toss was a 1 or 2. Find the probability mass functions for X and Y.

Homework 3 (due Wednesday 10/23, 5pm; 13 problems in all):    Reading: Chapter 4 Sections 4.9, 4.10, all of Chapter 5, and Chapter 10 through 10.2.1.
Ch.4 Problems, pp.180-182: 4.56, 4.67
Ch.4 Theoretical Problems, pp. 183-185: 4.10, 4.15
Ch.5 Problems, pp. 228-231: 5.6, 5.10, 5.23, 5.32, 5.36
Ch.5 Theoretical Problems, pp.232-233: 5.11, 5.25
plus 2 additional problems: (A). If U is a Uniform(0,1) random variable, find the density function of X = (U+3)2.
(B). (i) Find the expectation and 2nd moment of a Weibull(ν=0, α, β) random variable. (ii) Find the 4th moment of a Beta(2,3) random variable.

Homework 4 (due Monday 11/18, 5pm; 12 problems in all):    Reading: Chapter 6, omitting Sections 6.6 and 6.8, and Chapter 7 through Section 7.5.
Ch.6 Problems, pp.291-295: 6.1, 6.8, 6.17, 6.24, 6.32, 6.58, 6.63
Ch.6 Theoretical Exercises, pp.297-298: 6.27, 6.36
Ch.7 Problems, pp.378-382: 7.9, 7.30, 7.48.

Homework 5 (12 problems in all) will be due Monday, December 9, 5pm in class: Reading: Chapter 7, Section 7.3 and 7.5 through end, Chapter 8, and Simulation Handout which is the same as the second Handout pdf under handouts (3) below.
Ch.7, 4 Problems, pp.382-385: #7.50, 7.52, 7.66, 7.82
Ch.7, 3 Theoretical Exercises, pp.388-389: #7.27 (only in case X,Y have conitunous joint density), and 7.43
Ch.8, 3 problems, pp.424-426: 8.4, 8.9, 8.23
2 Extra problems Sim.1 and Sim.3 at end of Simulation Handout.

Handouts

(1.)   Practice (i.e., sample) problems for Test 1 on Wednesday, October 9, are posted here. Tests in this course are closed-book, so you should practice and commit to memory all formulas (or derivations) that you think you will need.

(2.)   Practice (i.e., sample) problems for Test 2 on Wednesday, November 20, are posted here. Tests in this course are closed-book, so you should practice and commit to memory all formulas (or derivations) that you think you will need. There is also a linked summary of topics for Test 2.

(3.) Here are two handouts here, respectively on Transformation of Random Variables and on Random Number Generation and Simulation. These topics are very important for concrete illustration of the course material, as they allow us to generate and interpret `artificial data' to illustrate the meaning of our Probability Limit Theorems (Law of Large Numbers, Central Limit Theorem). In addition, Simulation gives us an `experimental' avenue to calculate via artificial data probabilities which may be too difficult to figure analytically.

(4.) Here is a handout on Normal Approximation to Binomial Distribution containing a word-problem worked example, as well as some numerical examples of the quality of the normal approximation to the Binomial. A graph comparing the distribution function values of Binom(100,.3) with its approximating normal distribution N(30,21) can be found here.

(5.) Here is another handout, on Binomial approximation to Hypergeometric random variables, and the Poisson approximation to the Binomial. In addition, some simulated-data results are given to show that the expectations and probability mass functions behave as they should according to the relative-frequency interpretation of probabilities.

Important Dates